Estimation of a Bernoulli parameter p from imperfect trials

Imperfect Bernoulli trials arise when the outcome of a Bernoulli experiment is not known with certainty. In signal processing, we often need to estimate a probability of occurrence p of an event from imperfect Bernoulli trials. A typical example is the estimation of the probability of a signal being present in noisy data. In his famous essay, Bayes solved the same problem but for perfect trials. In this letter, a solution is provided for imperfect trials. It is shown that it includes Bayes' solution as a special case.